Question

ABC Is a right angled triangale at B. then value of sin( A+C) is

Answer 1

If in a triangle, ABC tan A + tan B + tan C is smaller than 0, then is the triangle acute angled, right angled or obtuse angled?

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3 ANSWERS



Rohan Ganguly, studied Science at Kendriya Vidyalaya No. 2, AFS Hindan

Answered May 16, 2017 · Author has 290answers and 100.7k answer views

The triangle is an obtuse angled triangle.

We know,

tan(A+B+C)=tanA+tanB+tanC−tanAtanBtanC1−tanAtanB−tanAtanC−tanBtanCtan⁡(A+B+C)=tan⁡A+tan⁡B+tan⁡C−tan⁡Atan⁡Btan⁡C1−tan⁡Atan⁡B−tan⁡Atan⁡C−tan⁡Btan⁡C

A+B+C=πA+B+C=π

⟹tan(A+B+C)=0⟹tan⁡(A+B+C)=0

⟹tanA+tanB+tanC=tanAtanBtanC⟹tan⁡A+tan⁡B+tan⁡C=tan⁡Atan⁡Btan⁡C

⟹tanAtanBtanC<0⟹tan⁡Atan⁡Btan⁡C<0

Now we know that the tantan function isnegative in the second quadrant, so for the above condition to be satisfied there has to be one angle >90o>90o. Hence we can conclude that the triangle is an obtuse one.

Answer 2

when one side of triangle at B in 90° so then other two angle is 45° each because right angle triangle have one side is 90° so the other two angle sum is 90 then the sum of all sides of triangle is 180°
and the each angle of two other angle is 90°/2=45°
thus A=45° and C=45°
so
sin(A+C)
=sinACosC+cosAsinC
=sin45°cos45°+cos45°sin45°
=1/√2×1/√2+1/√2×1/√2
=1/2+1/2
=1

so,sin(A+C)=1